Let V be a nite set of points in the Euclidean d-space (d 2).The intersection of all unit balls B(v, 1) centered at v, where v ranges over V , henceforth denoted by B(V ) is the ball polytope associated with V . After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes.(a) Dene the boundary complex of B(V ), i.e., dene its vertices, edges and facets in dimension 3, and investigate its basic properties. 2 (b) Apply results of this investigation to characterize nite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V . A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950's by means of ball polytopes, that the diameter of V is attained at most 2|V | − 2 times. Call V extremal if its diameter is attained this maximal number 2|V | − 2 of times. We extend the aforementioned result by showing that V is extremal i V coincides with the set of vertices of its ball polytope B(V ) and show that in this case the boundary complex of B(V ) is self-dual in some strong sense. The problem of constructing 2 This face structure turns out to be quite intricate or, quoting [2], p. 202, lines 2223: the face structure of these objects [ball polytopes] is not obvious at all.

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A convex geometric graph G of order n consists of the set of vertices of a plane convex n-gon P together with some edges and/or diagonals of P as edges. Call G l-free if G does not have l disjoint edges in convex position. We answer the following questions: (a) What is the maximum possible number of edges of G if G is l-free (as a function of n and 1)? (b) What is the minimum possible number of edges of G if G is l-free and saturated, i.e., if G U {e} is not l-free for any edge or diagonal e of P that is not already in G. We also fully describe the graphs G where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word "disjoint" from the definition of "l-free" and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turk's theorem, etc.

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